This simple model demonstrates a phase transition behavior emerging from the interactions among multiple agents in the presence of noise.

At a noise level higher than a certain threshold, the system generates symmetric behavior (e.g. vapor or melt of magnetization) or disagreement; whereas at a noise level lower than the threshold, the system exhibits spontaneous symmetry breaking (solid or magnetization) or consensus.

This provides some simple intuitions for understanding the impact of noise in communication networks on network behavior and the impact of noise (temperature) in phase transitions.

NOTE: For this simulator, the following terms are used interchangeably: Consensus, agreement, asymmetry, and all agents taking the same value.Disagreement, asymmetry, and approximately half of the agents taking +1 (colored as red) and the other half taking -1 (colored as blue).

## HOW IT WORKS

A group of non-mobile agents form a network, and each agent, taking bipolar value either +1 or -1, updates its value according to its previous value and the noisy measurements of the values of the agents connected to it.

Update rule: At each moment, each turtle receives all values from all turtles connected to it (including the turtle itself); these values are then averaged at the turtle and perturbed by a noise. The noise is uniformly distributed in interval [-z, z], where z is called the noise level. If the perturbed average value is positive, then the turtle will take value +1, otherwise it will take value -1.

In other words, each turtle first aggregates the opinions of all its connections and itself (subject to communication/observation noise perturbation), and then follows the aggregated opinion.

## HOW TO USE IT

To simulate, press SETUP to create the turtles and links, and then press GO to see the network evoluation or GO ONCE to go one step.

The following may be set before SETUP:

POPULATION: Use the slider to set the value for the number of turtles in the simulation. A very high POPULATION may slow down the simulation.

NOISE-LEVEL: Use the slider to set the noise level.

RANDOM-CONNECTIONS?: If ON is chosen, then the network forms a random graph process, with probability p (set by P-FOR-RANDOM-CONNECTIONS slider) to form a link between any two turtles. If OFF is chosen, then the network is over a fixed graph formed by connecting the nearest turtles within a distance set by VISION slider.

VISION: For the fixed graph case (RANDOM-CONNECTIONS?=OFF) only. Two turtles are connected by a link if they are within the distance set by VISION sider.

P-FOR-RANDOM-CONNECTIONS: For the random graph case (RANDOM-CONNECTIONS?=ON) only. For any turtle pair, the probablity of forming a connection (i.e. a link) is given by P-FOR-RANDOM-CONNECTIONS slider.

MIN/MAX NOISE LEVEL GUARANTEEING AGREEMENT: This displays, for a network after SETUP button is pressed, a pair of numbers such that if NOISE-LEVEL slider chooses any value between the min and max, then agreement is guaranteed.

The following are output:

\#RED, #BLUE: These display the number of red turtles (with state value +1) and the number of blue turtles (with state value -1) at the current time instant.

PREDCTION: This displays the prediction message. After SETUP button is preseed, the simulator makes a prediction of the behavior based on some analytic results. The prediction message may be: "Network will reach consensus (agreement/symmetry)!""Network will reach symmetry (disagreement)!""All connected sub-graphs will reach symmetry (disagreement)!""Network behavior cannot be predicted!"

SUM SEQUENCE: Each data point of the sequence corresponds to a tick in time and the sum of all turtles' state values at that time.

SUM OF SUM SEQUENCE: Each data point of the sequence corresponds to a tick in time and the accumulated sum of the SUM SEQUENCE by that time.

AVERAGE OF SUM SEQUENCE: Each data point of the sequence corresponds to a tick in time and the mean value of the SUM SEQUENCE by that time.

## THINGS TO NOTICE

If the noise level set by NOISE-LEVEL slider is higher than the displayed MAX NOISE LEVEL GUARANTEEING AGREEMENT, the system generates symmetric behavior (e.g. vapor or melt of magnetization) or disagreement; whereas at a noise level lower than MAX NOISE LEVEL GUARANTEEING AGREEMENT but above MIN NOISE LEVEL GUARANTEEING AGREEMENT, the system exhibits symmetry breaking (solid or magnetization) or consensus. This holds for both the fixed graph (RANDOM-CONNECTIONS?=OFF) and random graph process (RANDOM-CONNECTIONS?=ON).

If agreement is predicted, #RED, #BLUE, SUM SEQUENCE, SUM OF SUM SEQUENCE, and AVERAGE OF SUM SEQUENCE will reach/converge to some constants.

If disagreement is predicted, #RED, #BLUE, and SUM SEQUENCE will oscillate, SUM OF SUM SEQUENCE will exhibit a random walk behavior, and AVERAGE OF SUM SEQUENCE will tend to zero.